Then the discriminant $d$ of $f$ is $(2s)^2 - 4kn$, which by $(*)$ is $d = -4$. Observe that $H(-4) = 1$. There is only one reduced positive definite binary quadratic form with discriminant $-4$, namely:

(7)

\begin{align} \quad f(x, y) = x^2 + y^2 \end{align}

So we must have that $f \sim g$. So there exists integers $m_{11}, m_{12}, m_{21}, m_{22} \in \mathbb{Z}$ with $m_{11}m_{22} - m_{12}m_{21} = 1$ such that: